# Approximation by finite mixtures of continuous density functions that vanish at infinity

## TL;DR

This paper rigorously demonstrates that finite mixtures of continuous density functions vanishing at infinity can approximate a wide range of functions and densities in various modes, clarifying the scope of their approximation capabilities.

## Contribution

It provides formal proofs that finite mixtures of densities in  can approximate functions in multiple classes and modes, extending understanding of their approximation power.

## Key findings

- Finite mixtures can uniformly approximate functions in .
- Finite mixtures can approximate functions in  on compact sets.
- Finite mixtures can approximate functions in  in the  sense.

## Abstract

Given sufficiently many components, it is often cited that finite mixture models can approximate any other probability density function (pdf) to an arbitrary degree of accuracy. Unfortunately, the nature of this approximation result is often left unclear. We prove that finite mixture models constructed from pdfs in $\mathcal{C}_{0}$ can be used to conduct approximation of various classes of approximands in a number of different modes. That is, we prove approximands in $\mathcal{C}_{0}$ can be uniformly approximated, approximands in $\mathcal{C}_{b}$ can be uniformly approximated on compact sets, and approximands in $\mathcal{L}_{p}$ can be approximated with respect to the $\mathcal{L}_{p}$, for $p\in\left[1,\infty\right)$. Furthermore, we also prove that measurable functions can be approximated, almost everywhere.